Re: Coupled Diff Eqs or Poisson Eq, is symbolic solution

*To*: mathgroup at smc.vnet.net*Subject*: [mg102858] Re: [mg102844] Coupled Diff Eqs or Poisson Eq, is symbolic solution*From*: Murray Eisenberg <murray at math.umass.edu>*Date*: Mon, 31 Aug 2009 06:33:34 -0400 (EDT)*Organization*: Mathematics & Statistics, Univ. of Mass./Amherst*References*: <200908301006.GAA20836@smc.vnet.net>*Reply-to*: murray at math.umass.edu

Your post is essentially unreadable because of the embedded (hex?) codes such as "=E2", "=CF", etc. Please use plain ASCII and remove all such codes. Neelsonn wrote: > Guys, > > (This is the 3rd time I am trying to post this; I apologize for any > duplicate) > > I've have just installed Mathematica and have some tasks to accomplish > using it. I've spent some time trying to find similar problems at > Wolfram's website, but not success so far. So I am posting here for > the first time (unless someone, please, point me a similar post or > documentation) > > Here's what I need to solve: > (Poisson) > > =E2=88=87^2(x,y) = Rs * J(x,y) > > > for two cases: > > i) > > =CF=88 = 0 for x = 0, x = a, y = b; > > =E2=88=82=CF=88/=E2=88=82y = 0 for y = 0. > > and > > ii) > > =CF=88 = 0 for x = 0, x = a, y = 0, y = b. > > > Some side notes: > > - Physically speaking, for both cases I would like to know how the > electrostatic potential (=CF=88) will be distributed on a rectangular shape > (a,b) when it has grounded electrodes on the three edges (case i) and > grounded electrodes surrounding all four edges (case ii). > > - The rectangular shape resembles a resistive material, that comes the > Rs (sheet resistance) and J(x,y) is the current that is going to be > distributed on the surface of this geometry as well. In my case J(x,y) > = exp(V(x,y)). An "arrow plot" showing the current distribution will > be also interesting. > > - Eventually, once the solution =CF=88(x,y) is found, the electric field E > (x,y) = - =E2=88=87=CF=88 and the total current flow J = 1/=CF=81 * E, = > where =CF=81 is the > resistivity (ohm.meter), is also interested > > ---------- > > Now, my question is: Can Mathematica handle such problem and > boundaries like it is in order to solve it analytically (symbolic)? I > haven't seen, in the examples, problems like this. I wonder if I will > have to decouple > > =E2=88=87^2(x,y) = Rs * J(x,y) > > into first-order partial differential equations. Then, a follow-up > question that comes: can Mathematica do that automatically or I should > pose the problem myself? For that, I've seen an example from the > website that uses six first-order differential equations to solve the > kinetics of some chemical reactions. But the problem was solved > numerically and I would like to have an analytical equation as a > result. So is it possible to find such analytical solution in case I > have to use a system of first-order partial diff eqs? > > (I am pretty sure that this isn't a difficult problem for those who > Master Mathematica) > > A final question or better yet, help needed: I would like to do all > the above for a different shape, not a rectangule or square, but for a > trapezoidal shape. I have no idea how to start and don't know how the > boundaries will look like. I wonder if there is a way to draw such > shape in Mathematica and graphically tells the software to solve it > (like those "FEM softwares"...). That would be very easy! I would > really appreciate any input here. > > Thanks > N > > > > -- Murray Eisenberg murray at math.umass.edu Mathematics & Statistics Dept. Lederle Graduate Research Tower phone 413 549-1020 (H) University of Massachusetts 413 545-2859 (W) 710 North Pleasant Street fax 413 545-1801 Amherst, MA 01003-9305

**References**:**Coupled Diff Eqs or Poisson Eq, is symbolic solution possible?***From:*Neelsonn <neelsonn@gmail.com>